Who is Participating?

Yes,
I'd suggest you use postfix.
The process is you setup a mail server. Sendmail, postfix, qmail, exim, etc.
You configure the IP segment from which the clients will be connecting as allowed to relay. Different mail server have slightly different terminology for this functionality.
Now handling the outgoing based on the recipient domain, several have a transport routing setup
Commonly the format is
Recipient domain:nexthop
The default next hop might vary among the mail servers, before placing in production, test while looking at the maillog of the mail server ...
*:default next hop
:default next hop

The transport rule is only needed when the mail server does not have a direct path to the net. If it has a path to the net, you can allow it to use DNS to identify the servers responsible for each recipients domain and have it deliver the messages ....